**Subject:** The Bianchi Identity in ECE Theory

**Date:** Wed, 30 Apr 2008 05:35:30 EDT

The Bianchi identity as developed by Cartan is:

D ^ T := R ^ q

and as first shown in paper 15 (the most read ECE paper this month) is a rigorous identity consisting of a cyclic sum of three curvature tensors being identically the same as the cyclic sum of the fundamental definitions of these same three curvature tensors in terms of the gamma connection. Not carefully that the latter is more general than the Christoffel connection. This is proven in all detail in one of the appendices of paper 15. This proof is used again in paper 109, which proves rigorously the Hodge dual of the Bianchi identity:

D ^ T tilde := R tilde ^ q

These are in essence two expressions of the Bianchi identity, which is duality invariant. In paper 88 the rigorously correct expression of the so-called “second Bianchi identity” of the standard model was given. In paper 93 the Hodge dual identity was used to show the fatal internal inconsistency of the geometry of the EH equation. The latter geometry is:

T = 0, R ^ q = 0

whcih is fortuitously self consistent with the Bianchi identity, but the internal inconsistency is:

T tilde = 0, R tilde ^ q NOT zero in general

which is inconsistent with the Hodge dual of the Bianchi identity.

Paper 103 then produced a form of the field equation of realtivity in which torsion is accounted for correctly. Finally paper 110 shows that the torsion generator is the rotation generator within a factor of proportionality. The Bianchi identity is the basis of the homogeneous tensorial field equation of ECE, and the Hodge dual identity is the basis for the inhomogeneous tensorial field equation of ECE. These split into four vector equations. In dy namics these are the rigorously self-consistent “gravitomagnetic equations”, as they are obscurely known in the standard model, and in electrodynamics they are the rigorously self-consistent field equations, with a mathematical structure that is the same as the Maxwell Heaviside field equations, but written in a more general space-time with curvature and torsion both present, and with a non-zero connection.

ECE gives an internally consistent, duality invariant, structure for the field equations of dynamics and electrodynamics.

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